"[...] is it possible for this third party to lend my bitcoins [...]"

If the bitcoins are in your address and only you know the private key of your address, then nobody else can move your bitcoins away from your address.
However, if you deposit your bitcoins into a third party, theoretically they could do fractional reserve like normal banks do. This would cause an expansion of the currency issued by those third party, but not in the amount of bitcoins available in the blockchain.

"If so, is there any regulation or fractional reserve requirement for these institutions?"

The bitcoin territory is still very much unregulated. The judicial system is debating much broader issues than fractional reserve requirements. For example: there is still an on going debate about whether bitcoin should be consider money or not.

"And finally, can this expansion of the money supply affect the price of bitcoin?"

Although the bitcoin supply in the blockchain is limited to 21Mi, third parties can issue their own currency backed by bitcoins. In a free market, supply and demand would regulate the exchange rate among such currencies and bitcoin.
Depending on how this expansion of third parties currency take place, this could increase or decrease the bitcoin price.